Let $T$ be a complete $\mathcal L$-theory.
Monster models are essentially just used as expository devices in model theory when it is convenient to view all the models and sets involved in a discussion as being contained (Saturated Implies Universal) in one big model that realizes "lots" of types and has "lots" of automorphisms.
Consequently, such objects are discussed in fairly loose terms. The definition depends on the choice of $\kappa$ (though Saturated Models of same Cardinality are Isomorphic), and in some sense, the choice of $\kappa$ will depend on the models and sets that we intend to discuss. However, in practice, monster models are spoken about as though there is only one, and typically one is selected without mentioning $\kappa$, implicitly taking it to be strictly larger than the cardinality of any of the models and sets of parameters that are going to be discussed. Informally, this is justifiable since as long as the monster model is selected with $\kappa$ large enough, then anything that can be said about those substructures using those parameters will be true exactly when it is true in any larger monster model anyway (since even $\mathfrak C$ itself will be an elementary substructure of the larger monster model).